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A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Luisa Fattorusso (2008)

Czechoslovak Mathematical Journal

Let Ω be a bounded open subset of n , n > 2 . In Ω we deduce the global differentiability result u H 2 ( Ω , N ) for the solutions u H 1 ( Ω , n ) of the Dirichlet problem u - g H 0 1 ( Ω , N ) , - i D i a i ( x , u , D u ) = B 0 ( x , u , D u ) with controlled growth and nonlinearity q = 2 . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

A nonsmooth exponential

Esteban Andruchow (2003)

Studia Mathematica

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set s a of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator L ξ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), e x p ( ξ ) = e i L ξ , is continuous but not differentiable. The same holds for the Cayley transform C ( ξ ) = ( L ξ - i ) ( L ξ + i ) - 1 . We also show that the unitary group U L ² ( , τ ) with the strong operator topology is not an embedded submanifold...

Bayoumi Quasi-Differential is different from Fréchet-Differential

Aboubakr Bayoumi (2006)

Open Mathematics

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

Embedding of Hilbert manifolds with smooth boundary into semispaces of Hilbert spaces

J. Margalef-Roig, Enrique Outerelo-Domínguez (1994)

Archivum Mathematicum

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into H × [ 0 , + ) , where H is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in H × { 0 } . Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class with smooth boundary.

Extension of smooth functions in infinite dimensions II: manifolds

C. J. Atkin (2002)

Studia Mathematica

Let M be a separable C Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a C function, or of a C section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a C function on the whole of M.

Implicit functions from locally convex spaces to Banach spaces

Seppo Hiltunen (1999)

Studia Mathematica

We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller C Π k -map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.

Induced differential forms on manifolds of functions

Cornelia Vizman (2011)

Archivum Mathematicum

Differential forms on the Fréchet manifold ( S , M ) of smooth functions on a compact k -dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map Ω p ( M ) Ω p - k ( ( S , M ) ) (hat pairing with a constant function) and the bar map Ω p ( M ) Ω p ( ( S , M ) ) (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].

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